OFFSET
1,2
COMMENTS
This is the least n-parasitic number. A k-parasitic number (where 1 <= k <= 9) is one such that when it is multiplied by k, the product obtained is merely its rightmost digit transferred in front at the leftmost end.
Also sometimes called Dyson numbers or least n-transposable numbers. - Robert D. Rosales, Aug 11 2026
REFERENCES
C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford Univ. Press UK 2000.
LINKS
Wikipedia, Parasitic number.
P. Yiu, k-left-transposable integers, Chap.18.2 pp. 168/360 in 'Recreational Mathematics'
FORMULA
a(n) = min_{k=n..9}(k*(10^(ord(10,(10n-1)/gcd(k,10n-1)))-1)/(10n-1)). - Robert D. Rosales, Aug 11 2026
EXAMPLE
102564 is 4-parasitic because we have 102564*4=410256.
For n=5: 142857*5=714285. - Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009
MATHEMATICA
Table[With[{k = Range[n, 9]}, Min[k (10^(If[# == 1, 1, MultiplicativeOrder[10, #]] & /@ ((10 n - 1)/GCD[k, 10 n - 1])) - 1)/(10 n - 1)]], {n, 9}] (* Robert D. Rosales, Aug 11 2026 *)
PROG
(Python)
from sympy import n_order; from math import gcd
[min(k * (10**(n_order(10, m) if (m := (10*n - 1) // gcd(k, 10*n - 1)) > 1 else 1) - 1) // (10*n - 1) for k in range(n, 10)) for n in range(1, 10)] # Robert D. Rosales, Aug 11 2026
CROSSREFS
KEYWORD
fini,full,base,nonn
AUTHOR
Lekraj Beedassy, Aug 21 2004; corrected Dec 17 2004
EXTENSIONS
Edited by N. J. A. Sloane, Apr 13 2009
Corrected to set 5th term to 142857 as this is the least 5-parasitic number. Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009
a(9) added by Ian Duff, Jan 03 2012
Incorrect formula removed by Alois P. Heinz, Feb 18 2020
STATUS
approved